Determining properties of earth formations using the electromagnetic coupling tensor

ABSTRACT

A system and method to determine earth formation properties by positioning a logging tool within a wellbore in the earth formation, the logging tool having a tool rotation axis and a first, a second, and a third tilted transmitter coil, and a tilted receiver coil; rotating the logging tool about the tool rotation axis; energizing each transmitter coil; measuring a coupling signal between each transmitter coil and the receiver coil for a plurality of angles of rotation; determining a coupling tensor; and determining the earth formation properties using the coupling tensor.

CROSS-REFERENCE TO OTHER APPLICATIONS

This application is a continuation application and claims priority toand the benefit of U.S. Nonprovisional patent application Ser. No.11/610,653, filed Dec. 14, 2006 now U.S. Pat. No. 7,656,160, under 35U.S.C. §120.

FIELD OF THE INVENTION

The present invention relates to well logging and, more particularly, todetermining earth formation properties using the entire electromagneticcoupling tensor of an earth formation in a gain-corrected manner.

BACKGROUND

In petroleum exploration and development, formation evaluation is usedto determine whether a potential oil or gas field is commerciallyviable. One factor to determining the commercial viability of apotential field is the resistivity of the earth formation. Theresistance to electric current of the total formation—rock andfluids—around the borehole is the sum of the volumetric proportions ofmineral grains and conductive water-filled pore space. If the pores arepartially filled with gas or oil, which are resistant to the passage ofelectrical current, the bulk formation resistance is higher than forwater-filled pores.

Conventional induction logging tools use multiple coils to measure theconductivity (i.e., the inverse of resistivity) of the formation.However, formation conductivity is not a single number because theformations are invariably anisotropic, i.e., directionally dependent,which causes the conductivity to be a tensor quantity. As a result themore recent induction tools have been designed with multiple transmitteror receiver coils whose magnetic moments are in multiple directions andmeasurements between these coils are sensitive to more than onecomponent of the conductivity (or more generally, impedance) tensor.

For instance, in 3D array induction imager wireline tools (e.g.,3D-AIT™), both transmitter and receiver coils have magnetic dipolemoments in the x, y, and z directions, with z defined as along the axisof the tool. This is an improvement of conventional induction tooldesign, where only z-directed coils are employed. As an example,energizing the transmitter coil (T coil) in the x-direction andmeasuring with a receiver coil (R coil) that is in the y-directionprovides the xy-component of the coupling tensor in the tool frame ofreference. Other combinations of the transmitter and receiver coils canprovide remaining components of the coupling tensor and characterize theformation.

Similarly, LWD (logging-while-drilling) tools may be designed withreceiver antennas having magnetic dipole moments tilted relative to thez-direction and transmitter antennas having magnetic dipole momentsparallel to the z-direction. The tilted receivers may be, for example,in the z- and x-directions and provide measurements that are a linearcombination of those two signals. As the LWD tool rotates during normaldrilling operations, the tilted receivers sample formation properties inmultiple directions and can provide many, but not all, of the componentsof the coupling tensor. As the tool penetrates the earth, other earthlayers come within the depth of investigation of these measurements andthe distance to these boundaries can be extracted from the measurementsand used for geosteering purposes.

An inherent difficulty in using these tools is that the coil efficiency,and electronic drift affects the coupling between T and R coils. Thus,the T-R signal is not just a function of the medium filling the spacebetween the T and R antennas, and needs to be corrected for coilsensitivity and drift. With current designs, for example, there are noextra measurements available to enable one to estimate these couplings(e.g., gains) and one must assume that the gains remain constant anduphole measurements (e.g., calibrations) can be used to correct forthem.

In logging, the borehole compensation (BHC) technique provides a methodof self-calibrating electromagnetic measurements. BHC consists ofplacing two outer sensors symmetrically on the two sides of the centersensors. For four coils, made of two transmitters and two receivers, thecoil arrangements along the tool axis are either T1-R1-R2-T2 orR1-T1-T2-R2. By taking appropriate ratios of four basic andun-calibrated measurements, one can create a quantity that isindependent of coil gains. The method is based on taking two ratiosleading to Equation (P1) below:

$\begin{matrix}{\frac{\left( {T\; 1} \right)\left( {R\; 1} \right)S_{11}}{\left( {T\; 2} \right)\left( {R\; 1} \right)S_{21}}*\frac{\left( {T\; 2} \right)\left( {R\; 2} \right)S_{22}}{\left( {T\; 1} \right)\left( {R\; 2} \right)S_{12}}} & \left( {{{EQ}.\mspace{14mu} P}\; 1} \right)\end{matrix}$where the antenna efficiencies are shown in parentheses and S_(ij)represents the desired signal received from transmitter i by receiver j.

As can be seen, R1 is common in the first fraction and the gain of R1receiver cancels in taking the first ratio, the gain of R2 cancelstaking the second ratio, and when the two ratios are multipliedtogether, the gain of T1 and T2 cancels. The net result is a ratiomeasurement that, if expressed in logarithmic form, leads to anamplitude ratio and a phase shift, both of which are gain corrected. Inthis example, because the coils or sensors are aligned with the z-axis,only the zz-component of the measurement tensor is determined. Thismethod works well when the antennas are arranged symmetrically as inCDR™ (Compensated Dual Resistivity) and EPT™ (ElectromagneticPropagation Tool) devices.

For logging tools attempting to characterize the whole coupling tensor,the sign (or phase for complex quantities) of off-diagonal terms is veryimportant as it is used for log interpretation purposes. The BHC methodworks by taking ratios of the measurements, which introduces signambiguity. Examples include the ratio of two negative terms and anegative ratio in which it is not clear which term had a negative signoriginally. For LWD tools with receiver coils that are tilted relativeto the z-direction, the tool rotation may be used to generategain-corrected signal ratios. This technique partially solves theproblem, but limits the measurements to simple ratios of electromagneticcoupling tensor elements.

Therefore, it is a desire to provide a method of measuring the entirecoupling tensor, which is the preferred way of inferring earthconductivity anisotropy and the distance to boundaries separating mediaof different conductivities. It is a further desire to make thesemeasurements in a gain-corrected fashion with minimum requirements onthe hardware. The present invention proposes a solution to characterizeboth the coupling tensor and the gain corrections.

SUMMARY OF THE INVENTION

The invention comprises a system and method to determine earth formationproperties by positioning a logging tool within a wellbore in the earthformation, the logging tool having a tool rotation axis and a first, asecond, and a third tilted transmitter coil, and a tilted receiver coil;rotating the logging tool about the tool rotation axis; energizing eachtransmitter coil; measuring a coupling signal between each transmittercoil and the receiver coil for a plurality of angles of rotation;determining a coupling tensor; and determining the earth formationproperties using the coupling tensor.

The foregoing has outlined the features and technical advantages of thepresent invention in order that the detailed description of theinvention that follows may be better understood. Additional features andadvantages of the invention will be described hereinafter which form thesubject of the claims of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other features and aspects of the present inventionwill be best understood with reference to the following detaileddescription of specific embodiments of the invention, when read inconjunction with the accompanying drawings, wherein:

FIG. 1 is a schematic drawing of a logging tool of the presentinvention;

FIG. 2 is a flow diagram of an embodiment determining the gain-correctedcomponents of a coupling tensor in accordance with the presentinvention; and

FIG. 3 is a flow diagram of another embodiment determining thegain-corrected components of a coupling tensor in accordance with thepresent invention.

DETAILED DESCRIPTION

Refer now to the drawings wherein depicted elements are not necessarilyshown to scale and wherein like or similar elements are designated bythe same reference numeral through the several views.

As used herein, the terms “up” and “down”; “upper” and “lower”; andother like terms indicating relative positions to a given point orelement are utilized to more clearly describe some elements of theembodiments of the invention. Commonly, these terms relate to areference point as the surface from which drilling operations areinitiated as being the top point and the total depth of the well beingthe lowest point. In addition, the terms “coil” and “antenna” may beused interchangeably herein in both the description and the claims.

FIG. 1 is a schematic drawing of a logging system of the presentinvention, generally denoted by 5. Tool 10 may be lowered into borehole15 that has been created in earth formation 20. Tool 10 may be any toolsuitable for measuring the resistivity (or conductivity) of fog nation20, among other factors relevant to the field of oil field logging. Inaddition to electrical characteristics of the formation, tool 10 may beable to collect measurements concerning other formation parameters, suchas NMR (nuclear magnetic resonance) data, for example.

Tool 10 includes subs 30 and 40. Sub 30 includes at least three antennas35(a)-(c), placed in proximity to each other. Antennas 35 may be anytype of antenna suitable for a logging tool, including high resolutionantennas or multi-frequency antennas. Antennas 35 may each operate atone or more frequencies that correspond to one or more diameters ordepths of investigation (DOI). Antennas 35 are “tilted”, meaning theyhave magnetic dipole moments having non-zero components along thez-axis, i.e., the tool rotation axis, but not entirely aligned with thez-axis. They can be azimuthally rotated relative to one another and mayhave different tilt angles. Each antenna must be linearly independent ofthe others.

Sub 40 contains at least one antenna 45. The spacing between sub 30 andsub 40 may be selected to achieve optimal results depending on theapplication. Antenna 45 may be any antenna suitable for a logging tool,including high resolution antennas or multi-frequency antennas. Antenna45 may operate at one or more frequencies that correspond to one or morediameters or depths of investigation (DOI). Antenna 45 is a tilted (asdefined above) coil antenna and is tilted from the tool rotation axis byφ degrees, where φ is non-zero. The spacing and tilt of receiver 45 maybe selected to achieve optimal results depending on the application.

By reciprocity, sub 30 may operate as either a transmitter or receiversub in opposition to the operation of sub 40. In one embodiment, sub 30operates as the transmitter sub, and sub 40 acts as the receiver sub.Accordingly, antennas 35 a, 35 b, and 35 c operate as transmitters(alternatively referred to herein as “T1”, “T2” and “T3”, respectively)and antenna 45 operates as a receiver (alternatively referred to hereinas “R1”).

In addition to subs 30 and 40, tool 10 may contain a power supply,control and telemetry circuits, computer processors (or similarcircuitry for analysis of data and measurements), and other componentssuitable for an electromagnetic resistivity logging tool. Thesecomponents may be incorporated in tool 10 or located uphole in devicesor facilities on the surface.

The conductivity of a medium may be determined from an analysis of theelectromagnetic coupling tensor. We begin the analysis by consideringthe coupling between a transmitter and receiver in some medium. If thetransmitter and receiver are both coil antennas, each can beapproximated as a magnetic dipole having a magnetic moment thatrepresents the efficiency and orientation of the antenna. Thetransmitters can be operated one at a time or, if slightly differentfrequencies are used such that the signals can be discriminated, yet areclose enough in frequency to be considered a single frequency, thetransmitters can be operated simultaneously.

The voltage induced or the coupling signal at receiver coil 45, V_(TR),as a result of a current flowing in a transmitter coil, e.g., coil 35 a,35 b or 35 c, is given by a tensor equation, shown as Equation (1):

$\begin{matrix}{V_{TR} = {{\overset{\_}{m}}_{T}^{t} \cdot \overset{\_}{\overset{\_}{Z}} \cdot {\overset{\_}{m}}_{R}}} & \left( {{EQ}.\mspace{14mu} 1} \right)\end{matrix}$where Z is the coupling tensor characterizing the medium between thetransmitter and receiver coils and is given by Equation (2),

$\begin{matrix}{\overset{\_}{\overset{\_}{Z}} = \begin{pmatrix}({xx}) & ({xy}) & ({xz}) \\({yx}) & ({yy}) & ({yz}) \\({zx}) & ({zy}) & ({zz})\end{pmatrix}} & \left( {{EQ}.\mspace{14mu} 2} \right)\end{matrix}$The coupling tensor can be used to determine earth formation propertiessuch as the conductivity tensor.

The components of the tensor are expressed as (ij), representing theelementary coupling between a transmitter and a receiver, where thetransmitter is oriented along the i-direction, and the receiver isoriented along the j-direction of a Cartesian coordinate system. Ingeneral, all quantities are complex numbers and the transpose of thematrix is the trans-conjugate of the matrix. The vectors m_(T) and m_(R)represent the magnetic moments of the transmitter and receiver coils,respectively.

The coordinate systems used herein are Cartesian coordinate systems(orthogonal and unitary basis vectors) in which the z-axis is alignedwith the tool axis. Quantities identified with a double bar are matricesor tensors, and quantities identified with a single bar are vectors.

When Equation (1) is applied to conventional induction tools, such as anarray induction tool, then vectors m_(R) and m_(T) have only az-component, and the (zz) component of the coupling tensor isdetermined. If the receiver is tilted such that the components of themagnetic moment are along the z- and x-directions while the transmittermagnetic moment is along the z, then the measurement result is a linearcombination of (zx) and (zz), weighted by the relative orientation ofthe receiver antenna.

While Equation (1) is independent of coordinate system, one must committo a particular coordinate system in which to do the computations priorto performing the computations. Using multiple coordinate systemssimplifies the expression of various quantities, but requires coordinatetransformations be performed to express all quantities in a commonsystem before performing the actual computations. For example, themagnetic moments can be easily expressed as constant vectors in arotating coordinate system, but it is more convenient to use thecoupling tensor expressed in a fixed system. Thus, the magnetic momentsmust be transformed from the rotating system to the fixed one if thecomputations are done in the fixed frame. If F is the matrixtransforming a vector from the rotating frame to the fixed frame, onecan obtain the transformations shown in Equations (3a-c):

$\begin{matrix}\left. {\overset{\_}{m}}_{T}\rightarrow{\overset{\_}{\overset{\_}{F}} \cdot {\overset{\_}{m}}_{T}} \right. & \left( {{{EQ}.\mspace{14mu} 3}\; a} \right) \\\left. {\overset{\_}{m}}_{R}\rightarrow{\overset{\_}{\overset{\_}{F}} \cdot {\overset{\_}{m}}_{R}} \right. & \left( {{{EQ}.\mspace{14mu} 3}\; b} \right) \\\left. \overset{\_}{\overset{\_}{Z}}\rightarrow\overset{\_}{\overset{\_}{Z}} \right. & \left( {{{EQ}.\mspace{14mu} 3}\; c} \right)\end{matrix}$

This coordinate transformation is required, for example, to account forthe rotation of tool 10 during drilling. At some time, t, tool 10 isrotated around the tool axis (defined as the z-axis) by an angle θrelative to the fixed reference frame (the fixed frame being initiallyaligned with the rotating frame, but with its x-axis fixed relative tothe “top of hole” or magnetic north, for example). Here the rotation isclockwise as viewed from above. In this case the transforming matrix Fis in fact a rotation matrix, R, discussed below. For the purposes ofthe present disclosure, the rotating coordinate system shall be referredto as the “tool coordinate system”.

The transformation matrix from the tool coordinate system to the fixedsystem as described above is given by:

$\begin{matrix}{{\overset{\_}{\overset{\_}{R}}(\theta)} = \begin{pmatrix}{\cos(\theta)} & {- {\sin(\theta)}} & 0 \\{\sin(\theta)} & {\cos(\theta)} & 0 \\0 & 0 & 1\end{pmatrix}} & \left( {{EQ}.\mspace{14mu} 4} \right)\end{matrix}$The voltage at the receiver will become a function of the angle θ, eventhough the coupling tensor does not change.

Introducing Equation (4) into Equation (1) leads to:

$\begin{matrix}{\mspace{79mu}{{V_{TR}(\theta)} = {\left( {{\overset{\_}{\overset{\_}{R}}(\theta)} \cdot {\overset{\_}{m}}_{T}} \right)^{t} \cdot \overset{\_}{\overset{\_}{Z}} \cdot \left( {{\overset{\_}{\overset{\_}{R}}(\theta)}{\overset{\_}{m}}_{R}} \right)}}} & \left( {{{EQ}.\mspace{14mu} 5}\; a} \right) \\{\mspace{79mu}{{V_{TR}(\theta)} = {{\overset{\_}{m}}_{T}^{t}{{\overset{\_}{\overset{\_}{R}}(\theta)}^{t} \cdot \overset{\_}{\overset{\_}{Z}} \cdot {\overset{\_}{\overset{\_}{R}}(\theta)}}{\overset{\_}{m}}_{R}}}} & \left( {{{EQ}.\mspace{14mu} 5}\; b} \right) \\{\mspace{79mu}{{{V_{TR}(\theta)} = {{\overset{\_}{m}}_{T}^{t} \cdot {\overset{\_}{\overset{\_}{M}}(\theta)} \cdot {\overset{\_}{m}}_{R}}}\mspace{79mu}{{Wherein},}}} & \left( {{{EQ}.\mspace{14mu} 5}\; c} \right) \\{\mspace{79mu}{{{\overset{\_}{\overset{\_}{M}}(\theta)} \equiv {{\overset{\_}{\overset{\_}{R}}(\theta)}^{t} \cdot \overset{\_}{\overset{\_}{Z}} \cdot {\overset{\_}{\overset{\_}{R}}(\theta)}}}\mspace{79mu}{{with}\text{:}}}} & \left( {{{EQ}.\mspace{14mu} 5}\; d} \right) \\{{\overset{\_}{\overset{\_}{M}}(\theta)} = {\quad\begin{bmatrix}{\begin{matrix}{\frac{({xx}) + ({yy})}{2} +} \\\frac{({xy}) + ({yx})}{2}\end{matrix}\begin{matrix}{{\sin\left( {2\theta} \right)} +} \\\frac{({xx}) - ({yy})}{2}\end{matrix}{\cos\left( {2\theta} \right)}} & \cdots & {\begin{matrix}{\frac{({xy}) - ({yx})}{2} +} \\\frac{({yy}) - ({xx})}{2}\end{matrix}\begin{matrix}{{\sin\left( {2\theta} \right)} +} \\\frac{({xy}) + ({yx})}{2}\end{matrix}{\cos\left( {2\theta} \right)}} & \begin{matrix}{{({xz}){\cos(\theta)}} +} \\{({yz}){\sin(\theta)}}\end{matrix} \\{\begin{matrix}{\frac{({yx}) + ({xy})}{2} +} \\\frac{({yy}) + ({xx})}{2}\end{matrix}\begin{matrix}{{\sin\left( {2\theta} \right)} +} \\\frac{({xy}) - ({yx})}{2}\end{matrix}{\cos\left( {2\theta} \right)}} & \cdots & {\begin{matrix}{\frac{({xx}) + ({yy})}{2} -} \\\frac{({xy}) + ({yx})}{2}\end{matrix}\begin{matrix}{{\sin\left( {2\theta} \right)} -} \\\frac{({xx}) - ({yy})}{2}\end{matrix}{\cos\left( {2\theta} \right)}} & \begin{matrix}{{({yz}){\cos(\theta)}} -} \\{({xz}){\sin(\theta)}}\end{matrix} \\{{({zx}){\cos(\theta)}} + {({zy}){\sin(\theta)}}} & \cdots & {{({zy}){\cos(\theta)}} - {({zx}){\sin(\theta)}}} & ({zz})\end{bmatrix}}} & \left( {{EQ}.\mspace{14mu} 6} \right)\end{matrix}$

As shown in Equation (6), any component of the M(θ) matrix can berepresented as the sum of five possible terms, namely a constant term, asin(θ) term, a cos(θ) term, a sin(2 θ) term, and a cos(2 θ) term. Theorigin of these trigonometric terms is Equation (4) and itsmultiplication in Equation (5d), but these terms constitute thecomponents of the Fourier expansion of V_(TR)(θ). The coefficients canbe extracted if V_(TR)(θ) is measured for at least five differentangles, however there are nine components in the Z matrix and the systemis under determined.

Typically, instruments like AIT™ tools have all magnetic dipoles alignedwith the z-axis and accordingly measure a single complex quantityproportional to (zz), independent of θ. Recently developed LWD toolscontain receiver coils that are not aligned with the tool axis. Usingthe above method, these LWD tools may measure some, but not all, of thecomponents of the coupling tensor. With the implementation of these LWDtools, a limited number of gain-corrected elementary couplings areobtained.

The system and method of the present invention, however, obtains allcomponents of the coupling tensor in a gain-corrected fashion and with aminimum number of coils. The Fourier components in Equation (6) areweighted by the linear combination of the desired elementary couplings,namely (xy), (xx), etc. Finding those elementary couplings is a linearproblem.

Equation (6) may be expressed in a different form, as described below.The xx-component of the M(θ) matrix is given by:

$\begin{matrix}{{\overset{\_}{\overset{\_}{M}}(\theta)}_{xx} = {\frac{({xx}) + ({yy})}{2} + {\frac{({xy}) + ({yx})}{2}{\sin\left( {2\theta} \right)}} + {\frac{({xx}) - ({yy})}{2}{{\cos\left( {2\theta} \right)}.}}}} & \left( {{EQ}.\mspace{14mu} 8} \right)\end{matrix}$

Equation (8) contains three types of quantities: some elementarycouplings ((xx), (yy), (xy) and (yx)), some basis functions (sin(2θ) andcos(2θ)), and some constant coefficients. (+½ and −½). Because the finalobjective is to determine the values of the elementary couplings, it isadvantageous to separate the three sets of quantities into a vector ofbasis functions, F, a matrix of coefficients, M, and a vector ofelementary couplings, P such that M(θ)_(xx) can be expressed as shown inEquation (9):M (θ)_(xx) = F· M· P.  (EQ. 9)where, F and P are defined as:F =[1, sin(θ), cos(θ), sin(2θ), cos(2θ)];  (EQ. 10)P ^(t)[(xx),(yy),(zz),(xy),(xz),(yz),(yx),(zx),(zy)]; and  (EQ. 11M for the xx component is:

$\begin{matrix}{{\overset{\_}{\overset{\_}{M}}}_{xx} = {{1/2}*{\begin{bmatrix}1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\1 & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}.}}} & \left( {{EQ}.\mspace{14mu} 12} \right)\end{matrix}$

For the same transmitter-receiver antenna combination, M is always thesame, independent of the rotational angle or the properties of themedium, and depends only on the orientation of the antennas. F changesas a function of angle θ but is independent of the medium. Lastly, Pvaries as the measurements are done in different media.

Before introducing Equation (9) into Equation (5), the magnetic momentsof transmitter and receiver antennas, coils 35 a, 35 b, 35 c and 40,must be determined. These antennas may, in general, have threecomponents along the Cartesian coordinate system:

$\begin{matrix}{{{{\overset{\_}{m}}_{T} = \begin{pmatrix}\alpha_{1} \\\alpha_{2} \\\alpha_{3}\end{pmatrix}};}{and}} & \left( {{EQ}.\mspace{14mu} 13} \right) \\{{\overset{\_}{m}}_{R} = {\begin{pmatrix}\beta_{1} \\\beta_{2} \\\beta_{3}\end{pmatrix}.}} & \left( {{EQ}.\mspace{14mu} 14} \right)\end{matrix}$where the components contain the magnetic moment and the directioninformation.

Substituting Equations (9), (13), and (14) into Equation (5) leads toEquation (15):

$\begin{matrix}{{{V_{TR}(\theta)} = {\sum\limits_{i,j}{\alpha_{i}\beta_{j}{\overset{\_}{\overset{\_}{M}}(\theta)}_{ij}}}};{or}} & \left( {{{EQ}.\mspace{14mu} 15}a} \right) \\{{V_{TR}(\theta)} = {\sum\limits_{i,j}{\alpha_{i}\beta_{j}{\overset{\_}{F} \cdot {\overset{\_}{\overset{\_}{M}}}_{ij} \cdot \overset{\_}{P}}}}} & \left( {{{EQ}.\mspace{14mu} 15}b} \right) \\{{V_{TR}(\theta)} = {{\overset{\_}{F}(\theta)} \cdot \left( {\sum\limits_{i,j}{\alpha_{i}\beta_{j}{\overset{\_}{\overset{\_}{M}}}_{ij}}} \right) \cdot \overset{\_}{P}}} & \left( {{{EQ}.\mspace{14mu} 15}c} \right)\end{matrix}$

For the summation term in Equation (15), assuming that the gains of thetransmitter and receiver antennas are known, all the terms in the sumare known and are constant. Further, as shown in Equation (12), the sumis over matrices of (5×9) dimensions with appropriate weightingcoefficients, leading to a (5×9) final matrix, C, shown in Equation(16). Note matrix C depends only on coil orientation, i.e., tool layout.

$\begin{matrix}{{\overset{\_}{\overset{\_}{C}} = {\sum\limits_{i,j}{\alpha_{i}\beta_{j}{\overset{\_}{\overset{\_}{M}}}_{ij}}}};{and}} & \left( {{EQ}.\mspace{14mu} 16} \right) \\{{V_{TR}(\theta)} = {{\overset{\_}{F}(\theta)} \cdot \overset{\_}{\overset{\_}{C}} \cdot {\overset{\_}{P}.}}} & \left( {{EQ}.\mspace{14mu} 17} \right)\end{matrix}$

As noted above, for the same transmitter-receiver pair, the measurementshave to be done at a minimum of five angles. If Equation (17) isrepeated for k different angles (where k>=5), and the resultingequations are cast into matrix form, as shown in Equation (18), themeasured voltages turn into a k-dimensional vector and the F vectorbecomes a (k×5) matrix. The values for C and P remain unchanged.V _(TR)(θ)= F (θ)· C· P   (EQ. 18)

Although V_(TR) is a (k×1)-dimensional vector, the number of independentequations is determined by the rank of the matrix relation on the righthand side, which is only five. Thus, with measurements using onetransmitter and one receiver antenna, one can get a maximum of fiveindependent coupling components, which is not enough to characterize thecoupling tensor (or the P vector).

In order to determine all components of the P vector, more than one T-Rcombination is needed. Two different transmitter-receiver combinations,such as T₁R₁ and T₂R₁, for example, is also insufficient. With twotransmitter-receiver combinations, the rank of the combined matrix is,at most, equal to 8, which also does not provide enough measurements tocharacterize P.

Three transmitter-receiver combinations is sufficient, however. Becausetool 10 contains three different transmitters co-located, or placed inproximity with one other, and having a non-zero projection along thez-axis (e.g., coils 35), and the receiver (e.g., coil 45) is not alignedsolely with the z-axis, the combined matrix has a rank of 9:rank([ F _(T1,R) · C _(T1,R) · P );( F _(T2,R) · C _(T2,R) · P );( F_(T3,R) · C _(T3,R) · P )])=9=length(p)  (EQ. 19)where the notation on the left hand side is to show the three vectorsare concatenated.

Accordingly, with the configuration described above, it is possible tomap the measured voltages to all components of P. Any reciprocalmeasurement (e.g., swapping the role of transmitter and receiver) willgive equivalent properties, thus the transmitters and receivers can beoperated in a reciprocal manner. It is possible to have similar resultswith more coils, but it is preferable to minimize the number of requiredcoils in the system.

Gain is a complex multiplicative quantity, representing imperfections incoil manufacturing, and phase shift due to imperfect electronics, amongother factors. In one embodiment, a gain-corrected coupling tensor isobtained by assessing the relative gains between the transmitter andreceiver coils to provide accurate measurements.

We can describe the raw measurements of V_(TR)(θ), the relative gains,and all components of P within an unknown, complex, multiplicativefactor. Assuming that this multiplicative quantity is the absolute gainof the first transmitter-receiver combination, where (λ₂, λ₃) are therelative gains of the second and third coil combinations with respect tothe first transmitter-receiver combination, the system may be expressedas shown in Equations (20):

$\begin{matrix}\left\{ \begin{matrix}{{V_{{T\; 1},R}\left( \theta_{1} \right)} = {{\overset{\_}{\overset{\_}{F}}\left( \theta_{1} \right)} \cdot {\overset{\_}{\overset{\_}{C}}}_{{T\; 1},R} \cdot \overset{\_}{P}}} \\{{V_{{T\; 2},R}\left( \theta_{2} \right)} = {\lambda_{2}{{\overset{\_}{\overset{\_}{F}}\left( \theta_{2} \right)} \cdot {\overset{\_}{\overset{\_}{C}}}_{{T\; 2},R} \cdot \overset{\_}{P}}}} \\{{V_{{T\; 3},R}\left( \theta_{3} \right)} = {\lambda_{3}{{\overset{\_}{\overset{\_}{F}}\left( \theta_{3} \right)} \cdot {\overset{\_}{\overset{\_}{C}}}_{{T\; 3},R} \cdot \overset{\_}{P}}}}\end{matrix} \right. & \left( {{EQ}.\mspace{14mu} 20} \right)\end{matrix}$where θ_(j) is the set of angles at which the measurements fortransmitter-receiver j are recorded and used. The directions of thetransmitters are not the same, and as shown in Equation (16), the Cmatrices would be different.

(Ū₁, Ū₂ Ū₃) are defined as three vectors, such that:

$\begin{matrix}{\begin{bmatrix}\overset{\_}{U_{1}} \\{\overset{\_}{U_{2}}/\lambda_{2}} \\{\overset{\_}{U_{3}}/\lambda_{3}}\end{bmatrix} = {{\begin{bmatrix}{\overset{\_}{\overset{\_}{C}}}_{{T\; 1},R} \\{\overset{\_}{\overset{\_}{C}}}_{{T\; 2},R} \\{\overset{\_}{\overset{\_}{C}}}_{{T\; 3},R}\end{bmatrix}*\overset{\_}{P}} = {\overset{\_}{\overset{\_}{B}}*\overset{\_}{P}}}} & \left( {{EQ}.\mspace{14mu} 21} \right)\end{matrix}$where Ū₁ equals F ⁻¹(θ₁) V_(T1,R)(θ₁), and the other Ū_(i) are relatedsimilarly.

As mentioned above, C is a (5×9) matrix. As a result, B would be a supermatrix of (3×(5×9)) with a rank of 9. It follows that the kernel of Bhas six independent basis vectors (3×5−9). Those can be computedanalytically, simply by Gaussian elimination or preferably by applying asymbolic singular value decomposition (SVD) (QR factorization andbi-diagonalization) to get in both cases closed form expressions of thekernel basis. Nevertheless, it is sufficient to apply a numerical SVD onC and this will lead to exactly the same gains estimates as if closedform expressions are used. This computation has to be done once, as itonly depends on the layout of the tool. If SVD is chosen, one may notethat the last six rows of the 15*15 unitary matrix created span thedesired kernel. These vectors could also be computed using eigenvaluedecomposition of matrix B.

Let T be the matrix of size (6×15) composed of kernel basis vectors.Then, regardless of the value of P, T may be expressed as:

$\begin{matrix}{{\overset{\_}{\overset{\_}{T}}\begin{bmatrix}\overset{\_}{U_{1}} \\{\overset{\_}{U_{2}}/\lambda_{2}} \\{\overset{\_}{U_{3}}/\lambda_{3}}\end{bmatrix}} = 0} & \left( {{EQ}.\mspace{14mu} 22} \right)\end{matrix}$

T may be re-written in terms of unknown gains only:

$\begin{matrix}{{\overset{\_}{\overset{\_}{T}}\begin{bmatrix}\overset{\_}{U_{1}} \\{\overset{\_}{U_{2}}/\lambda_{2}} \\{\overset{\_}{U_{3}}/\lambda_{3}}\end{bmatrix}} = {{\overset{\_}{\overset{\_}{Q}} \cdot \begin{bmatrix}1 \\{1/\lambda_{2}} \\{1/\lambda_{3}}\end{bmatrix}} = 0}} & \left( {{EQ}.\mspace{14mu} 23} \right)\end{matrix}$

The system of Equation (23) is linear in (1/λ₂, 1/λ₃) and can be solvedin the least square sense leading to an unbiased estimate of the gains.The system will have a solution if the rank of the resulting matrix is2. Moreover, the equations involved in this linear system are relationsbetween DC components and second harmonics of Equation (6). If there isno second harmonic, the rank of Q will be less than 2. This is the casewhen, for example, the M(θ) matrix is diagonal with the same value for(xx) and (yy), corresponding to a homogeneous formation. Although thissituation would be rare, the accuracy of the gain estimation willdegenerate when the M(θ) matrix is close to being diagonal. In order toestimate gains in these cases, a second receiver 50 may be added closeto the first receiver 45. The ratios of the DC components measured bythis second receiver 50 will be estimates of (λ₂, λ₃).

After gains are estimated, they can be substituted into Equation (21) tocompute P by solving the system:

$\begin{matrix}{{\begin{bmatrix}{\overset{\_}{\overset{\_}{C}}}_{{T\; 1},R} \\{\overset{\_}{\overset{\_}{C}}}_{{T\; 2},R} \\{\overset{\_}{\overset{\_}{C}}}_{{T\; 3},R}\end{bmatrix}*\overset{\_}{P}} = \begin{bmatrix}\overset{\_}{U_{1}} \\{\overset{\_}{U_{2}}/\lambda_{2}} \\{\overset{\_}{U_{3}}/\lambda_{3}}\end{bmatrix}} & \left( {{EQ}.\mspace{14mu} 24} \right)\end{matrix}$

The resulting P components are gain corrected if the gain of the T1-Rantenna combination (e.g., coil 35 a and coil 45) is assumed equal tounity. To the extent that this gain is a scalar, complex quantity, thecomponents of the coupling tensor have been derived within a constantmultiplicative factor. If the actual gain of the first transmitter isknown, then λ₂ and λ₃ can be used to estimate the actual gains of theother transmitters. In this case, the constant multiplicative factor isknown.

The components of P, obtained above, are useful for estimating severalproperties, including vertical conductivity, horizontal conductivity,anisotropy, bed boundary locations, and orientation. However, it ispossible to construct some combination of these parameters that arewell-behaved and can be used as indicators without resorting to theinversion process. Symmetrized and anti-symmetrized combinations may beused, for example, for the operation of tool 10 and provide significantbenefits for geosteering.

In accordance with the system and method of the present invention, allcomponents of the Z matrix may be determined, making it possible tocreate any combination of elementary couplings. However, because theparameters are known within a multiplicative factor, it is advantageousto combine them in such a way that the multiplicative factor cancelsout. The following relation, for example, uses a general linearcombination of products of elementary components, Z_(ij), to some powerp_(ij), to construct a combined parameter, V, shown in Equation (25).

$\begin{matrix}{{V = \frac{\sum{w_{k}{\prod\;\left( Z_{i,j} \right)^{p_{i,j}}}}}{\sum{w_{k}^{\prime}{\prod\;\left( Z_{i,j} \right)^{p_{i,j}^{\prime}}}}}}{{{where}\mspace{14mu}{\sum p_{i,j}}} = \;{{\sum p_{i,j}^{\prime}} = {n.}}}} & \left( {{EQ}.\mspace{14mu} 25} \right)\end{matrix}$

The constants, w, are weighting coefficients. The total power, n, towhich the elementary components are raised is the same for the numeratorand the denominator, guaranteeing that the multiplicative gain factorcancels out.

As an example of the application of Equation (25), the followingcombined parameters, V₁ and V₂ are shown. For n=1:

$\begin{matrix}{{V_{1} = \frac{({xz}) - ({zx})}{({xx})}};{{{and}\mspace{14mu}{for}\mspace{14mu} n} = 2}} & \left( {{{EQ}.\mspace{14mu} 25}a} \right) \\{V_{2} = {\frac{{({xx})({yy})} - {({xy})({yx})}}{\left\lbrack {({xx}) + ({yy}) + ({zz})} \right\rbrack^{2}}.}} & \left( {{{EQ}.\mspace{14mu} 25}b} \right)\end{matrix}$It should be clear to one of ordinary skill in the relevant arts, thatany other combination of the elementary components, and any powers ofthem, can be used.

FIG. 2 shows a flow diagram illustrating a method of the presentinvention for measuring electromagnetic coupling between two toolsubsystems, each equipped with coils, generally denoted by the numeral100. At step 105, the logging tool is placed down a wellbore in aselected portion of the formation. As discussed above, in connectionwith FIG. 1, the logging tool contains two subs. The first sub, e.g., atransmitter sub, contains at least three coils. The second sub, e.g.,the receiver sub, contains at least one coil that is not aligned withthe tool axis. At step 110, the coils in the transmitter sub areenergized. In response, voltage is induced in the receiver coils at step115. The logging tool rotates about the tool axis in the wellbore as ittakes measurements of the formation, as shown in step 120. As discussedabove, the tool rotates at angle θ relative to a selected referencepoint.

At step 125, for the first transmitter-receiver pair, measurements aretaken of the voltage induced in the receiver coil as a result of thecurrent flowing in the first transmitter coil for k different angles,where k is greater than or equal to 5. As shown in steps 130 and 135,these measurements are repeated for the second and thirdtransmitter-receiver pairs. Accordingly, measurements of the voltageinduced in the receiver coil as a result of the current flowing in thesecond transmitter coil for at least five different angles are made asthe tool rotates. Similarly, the tool takes measurements of the voltageinduced in the receiver coil as a result of the current flowing in thethird transmitter coil for at least five different angles.

At step 140, all of the independent components of the coupling tensorare obtained. Next, at step 145, the coupling tensor is constructed.This complete measurement allows for the determination of earthconductivity anisotropy and the distance to boundaries separating mediaof different conductivities, among other electrical properties of theformation.

Although the coupling tensor constructed in step 145 can be used incertain applications, it is desirable to have a more accurategain-corrected coupling tensor. At step 150, the absolute gain for thefirst transmitter-receiver combination is obtained, as well as therelative gains of the second and third coil combinations with respect tothe first transmitter-receiver combination. Next, at step 155, thegain-corrected components of the vector of elementary couplings, i.e.,the P components, are obtained. The process for doing this is shown inEquations (20)-(24), discussed above. Although these gain-correctedcomponents are useful for determining a number of formation properties,it is also desirable to construct a combination of elementary couplings,as shown in step 160.

In another embodiment, a method for obtaining a gain-correctedelectromagnetic coupling tensor with closed form solutions is provided.The closed form solution can be used with three transmitters and onereceiver, as described above.

An example of the method is now described in relation to again-corrected electromagnetic coupling tensor with three transmittersand one receiver. The voltage induced at a receiver by anelectromagnetic field transmitted from a transmitter is given byEquation (5c) as shown above. Substituting Equation (6) into Equation(5c), we obtain Equation (26) which shows the voltage can be expressedin terms of a Fourier series of azimuth angle up to the second order.V _(TR)(φ)=C ₀ +C _(1c) cos(φ)+C _(1s) sin(φ)+C _(2c) cos(2φ)+C _(2s)sin(2φ);  (EQ. 26)where φ is the azimuth angle of the receiver, and we have defined a setof complex coefficients C₀, C_(1c), C_(1s), C_(2c), and C_(2s) torepresent the 0^(th), 1^(st), and 2^(nd) harmonics coefficients of thevoltage:

$\begin{matrix}{{C_{0} = \left\lbrack {{{zz}\;{\cos\left( \theta_{R} \right)}{\cos\left( \theta_{T} \right)}} + {\frac{1}{2}\left( {{xx} + {yy}} \right){\sin\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\cos\left( \phi_{T} \right)}} + {\frac{1}{2}\left( {{xy} - {yx}} \right){\sin\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\sin\left( \phi_{T} \right)}}} \right\rbrack};} & \left( {{{EQ}.\mspace{14mu} 27}a} \right) \\{{C_{1c} = \left\lbrack {{{xz}\;{\sin\left( \theta_{R} \right)}{\cos\left( \theta_{T} \right)}} + {{zx}\;{\cos\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\cos\left( \phi_{T} \right)}} + {{zy}\;{\cos\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\sin\left( \phi_{T} \right)}}} \right\rbrack};} & \left( {{{EQ}.\mspace{14mu} 27}b} \right) \\{{C_{1s} = \left\lbrack {{{yz}\;{\sin\left( \theta_{R} \right)}{\cos\left( \theta_{T} \right)}} + {{zy}\;{\cos\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\cos\left( \phi_{T} \right)}} - {{zx}\;{\cos\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\sin\left( \phi_{T} \right)}}} \right\rbrack};} & \left( {{{EQ}.\mspace{14mu} 27}c} \right) \\{{C_{2c} = \left\lbrack {{\frac{1}{2}\left( {{xx} - {yy}} \right){\sin\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\cos\left( \phi_{T} \right)}} + {\frac{1}{2}\left( {{xy} - {yx}} \right){\sin\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\sin\left( \phi_{T} \right)}}} \right\rbrack};} & \left( {{{EQ}.\mspace{14mu} 27}d} \right) \\{{C_{2s} = \left\lbrack {{\frac{1}{2}\left( {{xy} + {yx}} \right){\sin\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\cos\left( \phi_{T} \right)}} - {\frac{1}{2}\left( {{xx} - {yy}} \right){\sin\left( \theta_{R} \right)}{\sin\left( \theta_{T} \right)}{\sin\left( \phi_{T} \right)}}} \right\rbrack};} & \left( {{{EQ}.\mspace{14mu} 27}e} \right)\end{matrix}$where θ_(R) and θ_(T) are the receiver and transmitter angles withrespect to the tool axis, and φ_(T) is the azimuth angle of thetransmitter relative to the receiver.

The 0^(th) harmonic coefficient depends on the couplings zz, (xx+yy),and (xy−yx). The two 1^(st) harmonic coefficients depend on thecouplings xz, zx, yz, and zy; and the two 2^(nd) harmonic coefficientsdepend on (xx−yy) and (xy+yx). The harmonic coefficients can be dividedinto groups based on their coupling components: Group 1 includes the0^(th) harmonic coefficient; Group 2 includes the 1^(st) harmoniccoefficients; and Group 3 includes the 2^(nd) harmonic coefficients.(FIG. 3, step 310).

In step 315A of FIG. 3, the coupling components are determined from the2^(nd) harmonic coefficients:(xx−yy)=[C _(2c) cos(φ_(T))−C _(2s)sin(φ_(T))]/sin(θ_(R))sin(θ_(T));  (EQ. 28)(xy−yx)=[C _(2c) sin(φ_(T))+C _(2s)cos(φ_(T))]/sin(θ_(R))sin(θ_(T)).  (EQ. 29)In the above equations, the superscript i=1, 2, and 3 refers to thefirst, second, and third T-R pairs, respectively.

In step 315B, the relative gain of the second T-R pair with respect tothe first T-R pair can be calculated based on the ratio of (xx−yy) or(xy−yx) from the corresponding equations for those T-R pairs:

$\begin{matrix}\begin{matrix}{g_{2} = \frac{\left( {{xx}^{(2)} - {yy}^{(2)}} \right)}{\left( {{xx}^{(1)} - {yy}^{(1)}} \right)}} \\{{= \frac{\left\lbrack {{C_{2c}^{(2)}{\cos\left( \phi_{T}^{(2)} \right)}} - {C_{2s}^{(2)}{\sin\left( \phi_{T}^{(2)} \right)}}} \right\rbrack{\sin\left( \theta_{T}^{(1)} \right)}}{\left\lbrack {{C_{2c}^{(1)}{\cos\left( \phi_{T}^{(1)} \right)}} - {C_{2c}^{(1)}{\sin\left( \phi_{T}^{(1)} \right)}}} \right\rbrack{\sin\left( \theta_{T}^{(2)} \right)}}};{or}}\end{matrix} & \left( {{{EQ}.\mspace{14mu} 30}a} \right) \\\begin{matrix}{g_{2} = \frac{\left( {{xy}^{(2)} - {yx}^{(2)}} \right)}{\left( {{xy}^{(1)} - {yx}^{(1)}} \right)}} \\{{= \frac{\left\lbrack {{C_{2c}^{(2)}{\sin\left( \phi_{T}^{(2)} \right)}} + {C_{2s}^{(2)}{\cos\left( \phi_{T}^{(2)} \right)}}} \right\rbrack{\sin\left( \theta_{T}^{(1)} \right)}}{\left\lbrack {{C_{2c}^{(1)}{\sin\left( \phi_{T}^{(1)} \right)}} + {C_{2s}^{(1)}{\cos\left( \phi_{T}^{(1)} \right)}}} \right\rbrack{\sin\left( \theta_{T}^{(2)} \right)}}};}\end{matrix} & \left( {{{EQ}.\mspace{14mu} 30}b} \right)\end{matrix}$An average or weighted average from the two calculated relative gainvalues may be used to get a more accurate relative gain.

Similarly, the relative gain for the third T-R pair with respect to thefirst T-R pair can be calculated as:

$\begin{matrix}\begin{matrix}{g_{3} = \frac{\left( {{xx}^{(3)} - {yy}^{(3)}} \right)}{\left( {{xx}^{(1)} - {yy}^{(1)}} \right)}} \\{{= \frac{\left\lbrack {{C_{2c}^{(3)}{\cos\left( \phi_{T}^{(3)} \right)}} - {C_{2s}^{(3)}{\sin\left( \phi_{T}^{(3)} \right)}}} \right\rbrack{\sin\left( \theta_{T}^{(1)} \right)}}{\left\lbrack {{C_{2c}^{(1)}{\cos\left( \phi_{T}^{(1)} \right)}} - {C_{2c}^{(1)}{\sin\left( \phi_{T}^{(1)} \right)}}} \right\rbrack{\sin\left( \theta_{T}^{(3)} \right)}}};{or}}\end{matrix} & \left( {{{EQ}.\mspace{14mu} 30}c} \right) \\\begin{matrix}{g_{3} = \frac{\left( {{xy}^{(3)} - {yx}^{(3)}} \right)}{\left( {{xy}^{(1)} - {yx}^{(1)}} \right)}} \\{= {\frac{\left\lbrack {{C_{2c}^{(3)}{\sin\left( \phi_{T}^{(3)} \right)}} + {C_{2s}^{(3)}{\cos\left( \phi_{T}^{(3)} \right)}}} \right\rbrack{\sin\left( \theta_{T}^{(1)} \right)}}{\left\lbrack {{C_{2c}^{(1)}{\sin\left( \phi_{T}^{(1)} \right)}} + {C_{2c}^{(1)}{\cos\left( \phi_{T}^{(1)} \right)}}} \right\rbrack{\sin\left( \theta_{T}^{(3)} \right)}}.}}\end{matrix} & \left( {{{EQ}.\mspace{14mu} 30}d} \right)\end{matrix}$Note that the form of equations (30) comply with the more general resultof equation (24), but only exhibiting 4 equations instead of 6.

In step 320, the equations for the 1^(st) harmonic coefficients are usedto compute the couplings xz⁽¹⁾, zx⁽¹⁾, yz⁽¹⁾, and zy⁽¹⁾ using any two ofthe three transmitter-receiver pairs, as follows:

$\begin{matrix}{{{zx}^{(1)} = \frac{{\alpha\gamma}_{1} + \;{\beta\gamma}_{2}}{\alpha^{2} + \beta^{2}}};} & \left( {{EQ}.\mspace{14mu} 31} \right) \\{{{zy}^{(1)} = \frac{{\beta\gamma}_{1} - {\alpha\gamma}_{2}}{\alpha^{2} + \beta^{2}}};} & \left( {{EQ}.\mspace{14mu} 32} \right) \\{{{xz}^{(1)} = \frac{\begin{matrix}{C_{1s}^{(1)} - {{zx}^{(1)}{\cos\left( \theta_{R} \right)}{\sin\left( \theta_{T}^{(1)} \right)}}} \\{{\cos\left( \phi_{T}^{(1)} \right)} - {{zy}^{(1)}{\cos\left( \theta_{R} \right)}{\sin\left( \theta_{T}^{(1)} \right)}{\sin\left( \phi_{T}^{(1)} \right)}}}\end{matrix}}{{\sin\left( \theta_{R} \right)}{\cos\left( \theta_{T}^{(1)} \right)}}};} & \left( {{EQ}.\mspace{14mu} 33} \right) \\{{{yz}^{(1)} = \frac{\begin{matrix}{C_{1c}^{(1)} - {{zy}^{(1)}{\cos\left( \theta_{R} \right)}{\sin\left( \theta_{T}^{(1)} \right)}}} \\{{\cos\left( \phi_{T}^{(1)} \right)} + {{zx}^{(1)}{\cos\left( \theta_{R} \right)}{\sin\left( \theta_{T}^{(1)} \right)}{\sin\left( \phi_{T}^{(1)} \right)}}}\end{matrix}}{{\sin\left( \theta_{R} \right)}{\cos\left( \theta_{T}^{(1)} \right)}}};} & \left( {{EQ}.\mspace{14mu} 34} \right)\end{matrix}$where α, β, γ₁, and γ₂ are defined as:α=cos(θ_(R))[sin(θ_(T) ⁽¹⁾)cos(θ_(T) ⁽²⁾)sin(φ_(T) ⁽¹⁾)−sin(θ_(T)⁽²⁾)cos(θ_(T) ⁽¹⁾)sin(φ_(T) ⁽²⁾)];  (EQ. 35a)β=cos(θ_(R))[sin(θ_(T) ⁽²⁾)cos(θ_(T) ⁽¹⁾)sin(φ_(T) ⁽²⁾)−sin(θ_(T)⁽¹⁾)cos(θ_(T) ⁽²⁾)sin(φ_(T) ⁽¹⁾)];  (EQ. 35b)γ₁ =C _(1c) ⁽¹⁾ cos(θ_(T) ⁽²⁾)−C _(1c) ⁽²⁾ cos(θ_(T) ⁽¹⁾)/g ₂;  (EQ.35c)γ₂ =C _(1s) ⁽¹⁾ cos(θ_(T) ⁽²⁾)−C _(1s) ⁽²⁾ cos(θ_(T) ⁽¹⁾)/g ₂;  (EQ.35d)if the first and second T-R pairs are used, andα=cos(θ_(R))[sin(θ_(T) ⁽¹⁾)cos(θ_(T) ⁽³⁾)sin(φ_(T) ⁽¹⁾)−sin(θ_(T)⁽³⁾)cos(θ_(T) ⁽¹⁾)sin(φ_(T) ⁽³⁾)];  (EQ. 36a)β=cos(θ_(R))[sin(θ_(T) ⁽³⁾)cos(θ_(T) ⁽¹⁾)sin(φ_(T) ⁽³⁾)−sin(θ_(T)⁽¹⁾)cos(θ_(T) ⁽³⁾)sin(φ_(T) ⁽¹⁾)];  (EQ. 36b)γ₁ =C _(1c) ⁽¹⁾ cos(θ_(T) ⁽³⁾)−C _(1c) ⁽³⁾ cos(θ_(T) ⁽¹⁾)/g ₂;  (EQ.36c)γ₂ =C _(1s) ⁽¹⁾ cos(θ_(T) ⁽³⁾)−C _(1s) ⁽³⁾ cos(θ_(T) ⁽¹⁾)/g ₂;  (EQ.36d)if the first and third T-R pairs are used.

An average or weighted average of the two calculated sets of values forxz, zx, yz, and zy may be computed to get more accurate results.

In step 325, using a similar approach, the three C₀ equations for thethree T-R pairs are solved simultaneously using the relative gains foreach T-R pair. This leads to:

$\begin{matrix}{{\left( {{xx} + {yy}} \right) = \frac{{a_{22}\lambda_{1}} - {a_{12}\lambda_{2}}}{{a_{11}a_{22}} - {a_{12}a_{21}}}};} & \left( {{EQ}.\mspace{14mu} 37} \right) \\{{\left( {{xy} + {yx}} \right) = \frac{{a_{21}\lambda_{1}} - {a_{11}\lambda_{2}}}{{a_{11}a_{22}} - {a_{12}a_{21}}}};} & \left( {{EQ}.\mspace{14mu} 38} \right) \\{{{zz} = \frac{\begin{matrix}{C_{0}^{(1)} - {\frac{1}{2}\left( {{xx} + {yy}} \right){\sin\left( \theta_{R} \right)}{\sin\left( \theta_{T}^{(1)} \right)}}} \\{{\cos\left( \phi_{T}^{(1)} \right)} - {\frac{1}{2}\left( {{xy} - {yx}} \right){\sin\left( \theta_{R} \right)}{\sin\left( \theta_{T}^{(1)} \right)}{\sin\left( \phi_{T}^{(1)} \right)}}}\end{matrix}}{{\cos\left( \theta_{R} \right)}{\cos\left( \theta_{T}^{(1)} \right)}}};} & \left( {{EQ}.\mspace{14mu} 39} \right)\end{matrix}$where α₁₁, α₁₂, α₂₁, α₂₂, λ₁ and λ₂ are defined as:

$\begin{matrix}{{a_{11} = {\frac{1}{2}{{\sin\left( \theta_{R} \right)}\begin{bmatrix}{{\sin\left( \theta_{T}^{(1)} \right){\cos\left( \theta_{T}^{(2)} \right)}{\cos\left( \phi_{T}^{(1)} \right)}} -} \\{\sin\left( \theta_{T}^{(2)} \right){\cos\left( \theta_{T}^{(1)} \right)}{\cos\left( \phi_{T}^{(2)} \right)}}\end{bmatrix}}}};} & \left( {{{EQ}.\mspace{14mu} 40}a} \right) \\{{a_{12} = {\frac{1}{2}{{\sin\left( \theta_{R} \right)}\begin{bmatrix}{{\sin\left( \theta_{T}^{(1)} \right){\cos\left( \theta_{T}^{(2)} \right)}{\sin\left( \phi_{T}^{(1)} \right)}} -} \\{\sin\left( \theta_{T}^{(2)} \right){\cos\left( \theta_{T}^{(1)} \right)}{\sin\left( \phi_{T}^{(2)} \right)}}\end{bmatrix}}}};} & \left( {{{EQ}.\mspace{14mu} 40}b} \right) \\{{a_{21} = {\frac{1}{2}{{\sin\left( \theta_{R} \right)}\begin{bmatrix}{{\sin\left( \theta_{T}^{(1)} \right){\cos\left( \theta_{T}^{(3)} \right)}{\cos\left( \phi_{T}^{(1)} \right)}} -} \\{\sin\left( \theta_{T}^{(3)} \right){\cos\left( \theta_{T}^{(1)} \right)}{\cos\left( \phi_{T}^{(3)} \right)}}\end{bmatrix}}}};} & \left( {{{EQ}.\mspace{14mu} 40}c} \right) \\{{a_{22} = {\frac{1}{2}{{\sin\left( \theta_{R} \right)}\begin{bmatrix}{{\sin\left( \theta_{T}^{(1)} \right){\cos\left( \theta_{T}^{(3)} \right)}{\sin\left( \phi_{T}^{(1)} \right)}} -} \\{\sin\left( \theta_{T}^{(3)} \right){\cos\left( \theta_{T}^{(1)} \right)}{\sin\left( \phi_{T}^{(3)} \right)}}\end{bmatrix}}}};} & \left( {{{EQ}.\mspace{14mu} 40}d} \right) \\{{\lambda_{1} = {{C_{0}^{(1)}{\cos\left( \theta_{T}^{(2)} \right)}} - {C_{0}^{(2)}{{\cos\left( \theta_{T}^{(1)} \right)}/g_{2}}}}};} & \left( {{{EQ}.\mspace{14mu} 40}e} \right) \\{{\lambda_{2} = {{C_{0}^{(1)}{\cos\left( \theta_{T}^{(3)} \right)}} - {C_{0}^{(3)}{{\cos\left( \theta_{T}^{(1)} \right)}/g_{3}}}}};} & \left( {{{EQ}.\mspace{14mu} 40}f} \right)\end{matrix}$

Using the couplings zz⁽¹⁾, (xx⁽¹⁾+yy⁽¹⁾), (xy⁽¹⁾−yx⁽¹⁾), xz⁽¹⁾, zx⁽¹⁾,zy⁽¹⁾, yz⁽¹⁾, (xx⁽¹⁾−yy⁽¹⁾) and (xy⁽¹⁾+yx⁽¹⁾), the nine components ofthe coupling tensor for the first T-R pair may be obtained. The ninecomponents of the coupling tensor for the second and the third T-R pairsmay be obtained from those for the first T-R pair by multiplying thegain factors g₂ and g₃, respectively, from Eq. 30 (step 330).

From the foregoing detailed description of specific embodiments of theinvention, it should be apparent that a novel system and method todetermine the electromagnetic coupling tensor of an earth formation hasbeen disclosed. Although specific embodiments of the invention have beendisclosed herein in some detail, this has been done solely for thepurposes of describing various features and aspects of the invention,and is not intended to be limiting with respect to the scope of theinvention. It is contemplated that various substitutions, alterations,and/or modifications, including but not limited to those implementationvariations which may have been suggested herein, may be made to thedisclosed embodiments without departing from the scope of the inventionas defined by the appended claims.

What is claimed is:
 1. A method to determine earth formation properties,the method comprising: positioning a logging tool within a wellbore inthe earth formation, the logging tool having a first sub having at leastthree tilted transmitter antennas and a second sub having at least onetilted receiver antenna, wherein each of the antennas is linearlyindependent of the others; rotating the logging tool about a toollongitudinal axis; energizing the transmitter antennas; measuring acoupling signal between the transmitter antennas and the receiverantenna for a plurality of angles of rotation; determining components ofan electromagnetic coupling tensor; and determining the earth formationproperties using the electromagnetic coupling tensor components.
 2. Themethod of claim 1, wherein the energizing the transmitter antennascomprises energizing each transmitter antenna one at a time orenergizing all the transmitter antennas simultaneously using slightlydifferent frequencies.
 3. The method of claim 1, wherein the pluralityof angles of rotation comprises at least five angles for at least threetransmitter/receiver pairs.
 4. The method of claim 1, further comprisingoperating the transmitter antennas and the receiver antenna inreciprocal manners.
 5. The method of claim 1, wherein theelectromagnetic, coupling tensor components are gain-corrected.
 6. Themethod of claim 5, wherein the gain-corrected electromagnetic couplingtensor uses relative gains.
 7. The method of claim 1, wherein thelogging tool further comprises a second receiver antenna and thedetermining the electromagnetic coupling tensor components furthercomprises using the second receiver antenna to determine the relativegains.
 8. The method of claim 1, wherein the determining the earthformation properties comprises combining components of theelectromagnetic coupling tensor.
 9. The method of claim 1, wherein thedetermining the earth formation properties comprises determining aconductivity tensor.
 10. The method of claim 1, wherein the determiningthe components of an electromagnetic coupling tensor comprises using asingular value decomposition.
 11. The method of claim 1, wherein thedetermining the components of an electromagnetic coupling tensorcomprises using a Gaussian elimination.
 12. The method of claim 1,wherein the determining the components of an electromagnetic couplingtensor comprises using an eigenvalue decomposition.
 13. The method ofclaim 1, wherein the determining the components of an electromagneticcoupling tensor comprises using a closed-form solution.
 14. A systemused in a wellbore to determine earth formation properties, the systemcomprising: a logging tool disposed within the wellbore, the loggingtool rotating about a tool longitudinal axis; a first sub disposed onthe logging tool, the first sub having at least three tilted, linearlyindependent transmitter antennas; a second sub disposed on the loggingtool, the second sub having at least one tilted receiver antennalinearly independent from the three transmitter antennas;voltage-measuring circuitry disposed on the logging tool; and aprocessor to determine all of the electromagnetic coupling tensorcomponents and the earth formation properties using the electromagneticcoupling tensor components.
 15. The system of claim 14, wherein thelogging tool further comprises a second receiver antenna.
 16. The systemof claim 14, wherein at least two of the antennas are azimuthallyrotated relative to one another.
 17. The system of claim 14, wherein atleast two of the antennas have different tilt angles.
 18. The system ofclaim 14, wherein the subs are spaced apart from one another along thetool longitudinal axis.
 19. The system of claim 14, wherein at least oneof the antennas is a high resolution antenna.
 20. The system of claim14, wherein at least one of the antennas is a multi-frequency antenna.21. The system of claim 14, wherein the transmitter antennas areco-located or in close proximity to one another.